A Latin Square - Rocket Propellant *

A Latin Square is a one-factor design that extends blocking to two nuisance factors.  For example, suppose we want to study the effects of five different formulations of rocket propellant.  Suppose we also want to block (or average over) two nuisance factors:

1)    Batch of raw material

2)    Operator.

This is a 5 x 5 Latin Square and looks like this:

This arrangement is a square and with each of the five treatments (a formulation of rocket propellant) represented by a Latin letter.  Each letter occurs once for each operator and once for each batch of raw material.  The Latin Square allows blocking of two nuisance factors.  In this example we block on batch of raw material (the row) and operator (the column).

We will use this example to illustrate how to build and analyze a Latin square design using Design‑Expert^® software.

* Douglas Montgomery, Design and Analysis of Experiments, 4^th edition, John Wiley, Example 5-4 on page 196.

Rocket Propellant

Procedure to create a 5x5 Latin Square design:

1.    From the “File” menu choose “New Design…”

2.    On the “Factorial” tab choose “General Factorial” and “3” Categorical Factors:

3.    Name the first blocking factor (e.g. Row), enter the number of rows “5” and the block level names:

Rocket Propellant

4.    Name the second blocking factor (e.g. Column), enter the number of rows “5” and the block level names:

5.    Name the treatment factor (e.g. Formulation), enter the number of rows “5” and the factor level names:

Rocket Propellant

6.       

7.      

8.    Right click on the column heading “Std” and choose “Sort by Standard Order”.

9.    Right click on the column heading “Run” and choose “Re-order as currently displayed”.

Rocket Propellant

10.  Delete rows 26 – 125:

·        Left click on the button to the left of row 26.

·        While holding down the shift key, scroll down and left click the button to the left of the last row, i.e. row 125.

·        Right click on a button to the left of any row in the selected area and choose “Delete Row[s]”:

“Are you sure you want to delete these rows??”

“If you delete or ignore these rows you will have only
one level of factor/component 3.  Do you want to
delete/ignore these rows?”

Rocket Propellant

11.  Treatments (formulations) need to be assigned correctly.  Type in the Latin letters that represent the five treatments (formulations) in a cyclic order.  Starting at the top (row 1) type:

A, B, C, D, E     B, C, D, E, A     C, D, E, A, B     D, E, A, B, C

and finally  E, A, B, C, D

Rocket Propellant

12.     Right click on the column heading “Run” and choose “Randomize…”.

13.     Save your design as “Example 5-4.dx6”.

14.  Enter data.

·      Right click on the column heading "Std" and choose "Sort by Standard Order"

·      Type the following data into the response column. (You can copy / paste it if you are viewing this document online.)

15.     Analyze the response:

·        Click on the response node “Burn rate”.

·        Click on the “Effects” button.

a.     Right click on “A” and choose “Block”

b.    Right click on “B” and choose “Block”

c.     Right click on “C” and choose “Model”

Rocket Propellant

16.     Click on the “ANOVA” button:

17.     Click on the “Diagnostics” button and check as usual:

Rocket Propellant

18.     Note the Box-Cox plot recommends an inverse transformation, this makes sense for a rate.

19.     Reanalyze using the inverse transform:

Rocket Propellant

20.     Click on the “Model Graphs” button:

Plot the treatment (Formulation) and average over the blocks (Rows and Columns):